Finite Group Databasebeta

← #13086133

Subgroups — #13086133

Order 1440.

342 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#67150C211yesmaximal, characteristic
2C2×#67141C211yesmaximal, characteristic, complemented
3#10623829C211yesmaximal, characteristic, complemented
4Dic3×Dic15C211yesmaximal, characteristic, complemented
5C5×(C2×D3)⋊Dic3C211yesmaximal, characteristic, complemented
6C3×#6667C211yesmaximal, characteristic, complemented
7C3×(C2×D3)⋊Dic5C211yesmaximal, characteristic, complemented
8#91526—31nomaximal
9#91464—31nomaximal
10C22×C3×D15C2211yescharacteristic
11C22×C3×C5×D3C2211yescharacteristic
12C2×C3⋊Dic15C2211yescharacteristic
13C2×C5×C3⋊Dic3C2211yescharacteristic
14C2×C3×Dic15C2211yescharacteristic
15C2×C3×C5×Dic3C2211yescharacteristic
16C2×C32×Dic5C2211yescharacteristic
17#67150—22no—
18#67141—22no—
19#63913—51nomaximal, Hall
20C2×#1220—31no—
21C2×#1204—31no—
22C2×#1204—31no—
23C2×#1204—61no—
24C2×#1201—31no—
25C4×Dic15—31no—
26C5×#217—31no—
27C5×(C2×D3)⋊C4—31no—
28C3×C22⋊Dic5—31no—
29C3×#685—31no—
30#6667—31no—
31#6667D311yescharacteristic, complemented
32(C2×D3)⋊Dic5D311yescharacteristic, complemented
33#6665—31no—
34Dic3×Dic5—31no—
35C22×C32×C5C2311yescharacteristic
36C2×C3×D15—21no—
37C2×C3×D15—21no—
38C2×C3×D15—21no—
39C2×C3×C5×D3—21no—
40C2×C3×C5×D3—21no—
41C2×C3×C5×D3—21no—
42C3⋊Dic15D412yes—
43C5×C3⋊Dic3—21no—
44C3×Dic15—21no—
45C3×C5×Dic3—21no—
46C32×Dic5—21no—
47#4031—91noHall
48C2×#634—51no—
49C2×#625—51no—
50C3×(C2×D3)⋊C4—51no—
51C3×(C2×D3)⋊C4—51no—
52(C2×D3)⋊Dic3—51no—
53(C2×D3)⋊Dic3D511yescharacteristic, complemented
54Dic32—51no—
55C23×C3×C5—31no—
56C22×D15C2×D311yescharacteristic
57C22×C5×D3C2×D311yescharacteristic
58C22×C3×D5—31no—
59C2×C3×C4×C5—31no—
60#1220—62no—
61C2×Dic15—31no—
62C2×Dic15—31no—
63C2×Dic15—61no—
64C2×Dic15C2×D311yescharacteristic
65C2×C5×Dic3—31no—
66C2×C5×Dic3—31no—
67C2×C5×Dic3—61no—
68C2×C5×Dic3C2×D311yescharacteristic
69C2×C3×Dic5—21no—
70C2×C3×Dic5—31no—
71C2×C3×Dic5C2×D311yescharacteristic
72C2×C3×Dic5C2×D311yescharacteristic
73#1204—62no—
74#1204—62no—
75#1204—64no—
76#1201—62no—
77#885—151no—
78#885—151no—
79C2×C32×C5C2×D411yescharacteristic
80C2×C32×C5#4111yescharacteristic
81C2×C32×C5#4111yescharacteristic
82C3×D15—41no—
83C3×C5×D3—41no—
84C2×C22⋊D5—91no—
85C2×Dic10—91no—
86C5×C22⋊C4—91no—
87C22⋊Dic5—31no—
88#685—31no—
89#685—91no—
90C4×Dic5—91no—
91C22×C3×D3—51no—
92C22×C3×D3C2×D511yescharacteristic
93C2×C32×C4—51no—
94C2×C3⋊Dic3—51no—
95C2×C3⋊Dic3C2×D511yescharacteristic
96#634—102no—
97C2×C3×Dic3—51no—
98C2×C3×Dic3C2×D511yescharacteristic
99#625—102no—
100C22×C3×C5—21no—
101C22×C3×C5—61no—
102C22×C3×C5—61no—
103C22×C3×C5—61no—
104C22×C3×C5C22×D311yescharacteristic
105C22×C3×C5C22×D311yescharacteristic
106C2×D15—21no—
107C2×D15—21no—
108C2×D15—21no—
109C2×C5×D3—21no—
110C2×C5×D3—21no—
111C2×C5×D3—21no—
112C2×C3×D5—61no—
113C2×C3×D5—61no—
114C2×C3×D5—61no—
115C3×C4×C5—61no—
116Dic15—21no—
117Dic15—32no—
118Dic15—32no—
119Dic15—62no—
120C3×Dic5—22no—
121C3×Dic5—21no—
122C3×Dic5—21no—
123C3×Dic5—61no—
124C5×Dic3—21no—
125C5×Dic3—62no—
126C5×Dic3—61no—
127C5×Dic3—61no—
128C2×#67—151no—
129C2×#67—151no—
130C2×Dic6—151no—
131C2×Dic6—151no—
132C2×Dic6—301no—
133C3×C22⋊C4—151no—
134C3×C22⋊C4—151no—
135#217—31no—
136#217—151no—
137(C2×D3)⋊C4—31no—
138(C2×D3)⋊C4—51no—
139(C2×D3)⋊C4—51no—
140(C2×D3)⋊C4—151no—
141C4×Dic3—151no—
142C4×Dic3—151no—
143C32×C5#12411yescharacteristic, complemented, Hall
144C23×C5—31no—
145C22×D5—31no—
146C2×C4×C5—31no—
147C2×C4×C5—91no—
148C22⋊D5—182no—
149C2×Dic5—31no—
150C2×Dic5—91no—
151C2×Dic5D3211yescharacteristic
152Dic10—182no—
153C22×C32C22×D511yescharacteristic
154C2×C3×D3—21no—
155C2×C3×D3—21no—
156C2×C3×D3—21no—
157C2×C3×D3—101no—
158C2×C3×D3—101no—
159C2×C3×D3—101no—
160C32×C4—101no—
161C3⋊Dic3—21no—
162C3⋊Dic3—52no—
163C3×Dic3—21no—
164C3×Dic3—101no—
165#124—451noSylow
166C2×C3×C5—21no—
167C2×C3×C5—21no—
168C2×C3×C5—21no—
169C2×C3×C5—121no—
170C2×C3×C5#23511yescharacteristic
171C2×C3×C5#23511yescharacteristic
172C2×C3×C5D3×D411yescharacteristic
173C2×C3×C5D3×D411yescharacteristic
174C2×C3×C5#23711yescharacteristic
175C2×C3×C5#23711yescharacteristic
176D15—41no—
177C3×D5—121no—
178C5×D3—41no—
179C23×C3—31no—
180C23×C3—151no—
181C22×D3—51no—
182C22×D3D3×D511yescharacteristic
183C2×C3×C4—31no—
184C2×C3×C4—51no—
185C2×C3×C4—51no—
186C2×C3×C4—101no—
187C2×C3×C4—151no—
188#67—302no—
189#67—302no—
190C2×Dic3—31no—
191C2×Dic3—31no—
192C2×Dic3—51no—
193C2×Dic3—61no—
194C2×Dic3—151no—
195C2×Dic3—151no—
196C2×Dic3—301no—
197C2×Dic3D3×D511yescharacteristic
198Dic6—302no—
199Dic6—302no—
200Dic6—304no—
201C22×C5—61no—
202C22×C5—61no—
203C22×C5—61no—
204C22×C5C2×D3211yescharacteristic
205C2×D5—61no—
206C2×D5—61no—
207C2×D5—61no—
208C4×C5—61no—
209C4×C5—181no—
210Dic5—21no—
211Dic5—61no—
212Dic5—92no—
213C2×C32#70911yescharacteristic
214C2×C32D4×D511yescharacteristic
215C2×C32#71111yescharacteristic
216C3×D3—41no—
217C3×D3—201no—
218C2×Q8—451no—
219C2×D4—451no—
220C22⋊C4—91no—
221C22⋊C4—151no—
222C22⋊C4—151no—
223C22⋊C4—451no—
224C42—451no—
225C3×C5—21no—
226C3×C5#88511yescharacteristic, complemented
227C3×C5#88511yescharacteristic, complemented
228C22×C3—21no—
229C22×C3—61no—
230C22×C3—61no—
231C22×C3—61no—
232C22×C3—301no—
233C22×C3—301no—
234C22×C3—301no—
235C22×C3C2×D3×D511yescharacteristic
236C22×C3C2×D3×D511yescharacteristic
237C2×D3—21no—
238C2×D3—21no—
239C2×D3—21no—
240C2×D3—101no—
241C2×D3—101no—
242C2×D3—101no—
243C3×C4—61no—
244C3×C4—102no—
245C3×C4—101no—
246C3×C4—101no—
247C3×C4—301no—
248Dic3—21no—
249Dic3—62no—
250Dic3—61no—
251Dic3—61no—
252Dic3—101no—
253Dic3—152no—
254Dic3—152no—
255Dic3—302no—
256C2×C5—121no—
257C2×C5#379511yescharacteristic
258C2×C5#379511yescharacteristic
259C2×C5#381111yescharacteristic
260D5—121no—
261C32#403111yescharacteristic, complemented, Sylow
262C23—31no—
263C23—151no—
264Q8—902no—
265D4—902no—
266C2×C4—31no—
267C2×C4—51no—
268C2×C4—91no—
269C2×C4—151no—
270C2×C4—451no—
271C2×C3—21no—
272C2×C3—21no—
273C2×C3—21no—
274C2×C3—121no—
275C2×C3—601no—
276C2×C3#677111yescharacteristic
277C2×C3#677711yescharacteristic
278C2×C3#677911yescharacteristic
279C2×C3#678011yescharacteristic
280C2×C3#678211yescharacteristic
281C2×C3#679011yescharacteristic
282D3—41no—
283D3—201no—
284C5#6391311yescharacteristic, complemented, Sylow
285C22—61no—
286C22—61no—
287C22—61no—
288C22—301no—
289C22—301no—
290C22—301no—
291C22D32×D511yescharacteristic
292C4—61no—
293C4—101no—
294C4—181no—
295C4—301no—
296C4—452no—
297C3—21no—
298C3#9146411yescharacteristic, complemented
299C3#9152611yescharacteristic, complemented
300C2—121no—
301C2—601no—
302C2#1062403011yescharacteristic
303C2#1062405011yescharacteristic
304C2#1062405311yescharacteristic
subgroupG/H
center, Frattini, hypercenterC22D32×D5
commutator, Fitting, socle, cosocleC22×C32×C5C23
prime radicalC2×#67150C2
hypocenterC32×C5#124
LCSG ▷ C22×C32×C5 ▷ C32×C5
UCSC1 ◁ C22

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#67141C2semidirect
3. #10623829C2semidirect
4. Dic3×Dic15C2semidirect
5. C5×(C2×D3)⋊Dic3C2semidirect
6. C3×#6667C2semidirect
7. C3×(C2×D3)⋊Dic5C2semidirect
31. #6667D3semidirect
32. (C2×D3)⋊Dic5D3semidirect
53. (C2×D3)⋊Dic3D5semidirect
143. C32×C5#124semidirect
226. C3×C5#885semidirect
227. C3×C5#885semidirect
261. C32#4031semidirect
284. C5#63913semidirect
298. C3#91464semidirect
299. C3#91526semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#671502
C2×#671412
#106238292
Dic3×Dic152
C5×(C2×D3)⋊Dic32
C3×#66672
C3×(C2×D3)⋊Dic52
#915263
#914643
#639135